Final answer:
To find the equation of a plane that passes through a given point and is perpendicular to two given planes, we find the cross product of the normal vectors of the given planes and substitute the coordinates of the point into the equation of the plane.
Step-by-step explanation:
To find an equation of the plane that passes through the point (1, 5, 1) and is perpendicular to the planes 5x + y - 5z = 5 and x + 8z = 7, we need to find the cross product of the normal vectors of the given planes. The normal vector of the first plane is (5, 1, -5) and the normal vector of the second plane is (1, 0, 8). Taking the cross product of these two vectors gives us (-41, -37, -5).
Next, we substitute the coordinates of the given point (1, 5, 1) into the equation of the plane to find the constant term. Plugging in the values, we get -41(1) - 37(5) - 5(1) = -41 - 185 - 5 = -231.
Therefore, the equation of the plane that passes through the point (1, 5, 1) and is perpendicular to the planes 5x + y - 5z = 5 and x + 8z = 7 is -41x - 37y - 5z = -231.